Scaling properties of disordered multifractals

PHYSlCA ELSEVIER

Physica A 226 (1996) 34-63

Scaling properties of disordered multifractals Antoine Saucier lnstitutt for energiteknikk, P.O. Box 40, N-2007 Kjeller, Norway

Received 1 July 1994; revised 15 September 1995

Abstract

In turbulence, the simplest phenomenological models of the energy cascade are multiplicative processes constructed on a regular grid (in short, M.P.G). They have been used mostly for their simplicity, allowing many of their properties to be derived analytically, and their capacity to reproduce the scale invariance properties of various geophysical fields. However, these M.P.G.'s suffer from the drawback of lacking translation invariance in their spatial statistics (spatial homogeneity), and therefore they cannot be fully satisfactory models for geophysical fields. In this paper, we are interested in finding new construction methods for spatially homogeneous random multifractals. We investigate the scaling properties of a new family of gridless models of multifractals. P A C S : 47.25Cg; 42.20Tg; 02.50. +s; 05.40. +j; 64.60.Ak

1. Introduction

In fully developed turbulence, the simplest phenomenological models of the energy cascade are multiplicative processes constructed on a regular grid (denoted by M.P.G. in the following). These models, also called cascade processes, were introduced for the modelling of the energy dissipation field by Yaglom [1,2] and further studied by Novikov [ 3 - 7 ] and B. Mandelbrot ([8, 9] and later publications), and also by m a n y others such as Evertsz and Mandelbrot [9], Kahane and Peyri~re [10], Lavall~e et al. [11], Marshak and Davis [12, 13], Saucier [14 18], Schertzer and Lovejoy [19,20], Meneveau and Sreenivasan [21,22]. Since the early work of Yaglom, the use of cascade processes has been extended to the modelling of other irregular geophysical fields such as rain (radar reflectivity), clouds, porous media etc. In the context of turbulence, where M.P.G.'s are a model of the energy transfer from large to small scales, the process is thought to simulate an energy flux from a "mother eddy" to "daughter eddies". M.P.G.'s occurs on a regular grid and consequently daughter eddies are always contained inside their mother eddy (Fig. 1, top). For some 0378-4371/96/$15.00 ~2) 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 3 7 1 ( 9 5 ) 0 0 3 3 8 - X

A. Saucier/Physica A 226 (1996) 34- 63

35

simple non-linear dynamical systems, e.g. quadratic maps (Halsey et al. [23]), it is known that the invariant probability measure is described exactly by a multiplicative process (although there may not be a regular grid!). In contrast, one must recognize that for turbulent flows the discrete splitting of eddies characteristic of cascade models has not so far received a rigorous dynamical basis. It is rather unphysical to assume that the energy cascade proceeds dynamically on a regular grid, and indeed other phenomenological models such as the /~-model [26] do not make such assumptions. In spite of this problem, M.P.G.'s have been studied lbr their simplicity, that allows m a n y of their properties to be derived analytically, and their capacity to reproduce the scale invariance properties of many geophysical fields. The grid involved in a M.P.G. breaks the spatial homogeneity 1 of the field. This lack of translation invariance is usually not desirable for the modelling of geophysical fields. In two dimensions, this property causes the field to look like an airplane view of Manhattan, i.e. blocks of different sizes are visible and the grid underlying the construction remains perceptible (Fig. 4). Another unavoidable effect of the grid is the introduction of a privileged scale ratio fi, since the length scales involved in the process take the form ). "(in figure 1 the eddies have a size 1, 1/2, 1/4 and therefore ). = 2). In this paper, we are interested in finding new construction methods for spatially homogeneous multifractals. To address this problem, we shall investigate a new family of models that are constructed by exponentiating a sum of pulses of various sizes and amplitudes. By contrast with multifractals constructed on a grid, the pulse locations are random and mutually independent. In that sense, these multifractals are more "disordered". The absence of a grid in their construction also insures spatial homogeneity. For such disordered multifractals, the spatial positions of large and smaller scale eddies are random and independent (Fig. 1, bottom), and in that sense the relationship between the positions of mother and daughter eddies has been broken. The general goal of this paper is to study the effect of this randomisation on the scaling properties of the field. Schertzer and Lovejoy [19] have also introduced a different family of gridless multifractals called continuous cascades. In their model, sine and cosine functions with random phases play the role of pulses. In contrast with our model, where pulses are localised in real space, the pulses used in continuous cascades are de-localised in real space but localised in Fourier space. Several ingredients must be specified for our disordered multifractals: a set of length scales [11,12 . . . . . IN} representing the pulse sizes, the n u m b e r / ~ ( 1 , ) of pulses of size l,, and an amplitude factor ~(l,). The amplitude of the ruth pulse of size l,, takes the

1This terminology is borrowed from turbulence, where one defines "locally homogeneous turbulence" as a state of the velocityfield for which all statistics are invariant under translation within a restricted spatial domain. For example, if F(x) is a spatially homogeneous random function, then the correlation fimction (F(x)F(x + L)) is independent of x, and therefore depends only on L.

36

A. Saucier/Physica A 226 (1996) 34 63

Fig. 1. Top: a multiplicative process on a grid. The "eddies" are represented by circles with an arrow. The largest eddy of size 1 (the darkest) splits into four daughter eddies of size 1/2 (in grey), which in turn split into four "grand-daughter" eddies of size 1/4 (in white). It should be noted that daughter eddies are always contained inside their mother eddy, because of the presence of the grid underlying the process (indicated with dotted lines). Bottom: An illustration of the construction of a disordered multifractal. In contrast with the above cascade process on a grid (top), the positions of the eddies are mutually independent. The field is constructed by exponentiating this sum of randomly located pulses (the pulses, or "eddies", are cylindrical and have random amplitudes).

f o r m 8 ( l , ) U , , , , , w h e r e the r a n d o m v a r i a b l e s U,,,, are m u t u a l l y i n d e p e n d e n t a n d i d e n t i c a l l y d i s t r i b u t e d . O u r first g o a l is to d e t e r m i n e the c o n s t r a i n t s to be a p p l i e d o n the f u n c t i o n s M ( / , ) a n d ci(l,) to o b t a i n a m u l t i f r a c t a l field, a n d to u n d e r s t a n d h o w these c o n s t r a i n t s d e p e n d o n the set of l e n g t h scales {/1,12 . . . . , IN }. O u r s e c o n d goal is to find the scaling b e h a v i o u r of these d i s o r d e r e d m u l t i f r a c t a l s . F i n a l l y , we will also e x a m i n e the p o w e r s p e c t r u m of the l o g a r i t h m of the m u l t i f r a c t a l field to see if it is a 1 / f noise, as it is a l w a y s the case for c o n t i n u o u s cascades [19].

37

A. Saucier/Physica A 226 (1996) 34 63

2. Multifractals constructed on grids: brief review 2.1. Definitions In this section we focus mostly on r a n d o m multiplicative processes in one dimension. The construction process of the r a n d o m function ~:N(X) is illustrated in Fig. 2. In the first step of construction, the unit interval [0, 1] is split in ). sub-intervals li ( i = 1,2 . . . . . )3 of size l~ = ) - 1 and in each interval the function ~'~(x) is given a r a n d o m value W/, i.e. for each x e l i we have ~:l(x)= Wi. The W~ are positive mutually independent and identically distributed r a n d o m variables with unit expectation value, i.e. such that ~ ~ W for all i and ( W ) = 1 (the symbol ~ denotes an equality in probability distribution and the brackets ( . . . ) denote an ensemble average (i.e. expectation value)). In the second step of construction, each interval li is split in ). subintervals I/.~ (j = 1,2 . . . . . ,:3 of size / 2 = ,;. 2 and in the interval I~. i the value of the r a n d o m function e2(x)is defined to be W i W i, where Wv is independent of W,. and all other multipliers involved in the process. After iterating this process N times (N >~ 1) in a self-similar fashion, we obtain a r a n d o m function ~;~.(x) that is piecewise constant on a collection of 2 N intervals of size IN = )~-N. The value of the function on one of these intervals can always be written in the form C,N(X) - W l ( x ) W 2 ( x ) . . .

(1)

WN(X),

where W,(x) denotes the multiplier at scale I, = 2 - " corresponding to the location x. Since the multipliers are independent r a n d o m variables, it follows from Eq. (1) that ([~:N(X)] q) = ( w q ) N .

(2a)

Notice that the normalisation condition ( W ) = 1 implies via (2a) that (~:N(X)) = 1. Eliminating N from Eq. (2a) by using IN = 2-'~, Eq. (2a) can be rewritten in the form ( [~:N(X)] q ) = I~ K~('° ,

(2b)

I

I Wl(I) /

.W~(2)

/

I

Wlll) W2(l) / ,/

I

W2(2)

W1(2) W2(3)

I W2(4)

II W1(1)W2(1) II Wl(1)W2(2) • W1(2)W2(3) "W1(2)W2(4) I etc

. . .

Fig. 2. An illustration of the Mandelbrot canonical cascade in one dimension and with a splitting factor ,;. = 2. The weights W,.(j) are identically distributed and independent random variables.

A. Saucier/Physica A 226 (1996) 34 63

38

where KG(q) = log~ { ( W q ) }

(2c)

is called the generator of the cascade process. It follows from Eq. (2c) and the normalisation condition ( W ) = 1 that KG(1)= 0. The generator therefore satisfies Kc(0) = KG(1) = 0.

(2d)

It will be useful to write a more precise expression of the random function eN(x). As seen from Eq. (1), eu(x) has a multiplicative structure and therefore ln{cN(x)} has an additive structure, i.e. it can be regarded as a sum of functions of different amplitudes. More precisely, we can always write eu(x) in the form ~N(x) = exp

~ n=l

ln{W..m} O[(x - x.,m)/l.]

,

(3)

m=l

where x,,.m = l,,/2 + (m - 1)/. is the center of the ruth interval I.,,. = [(m - 1)/., ml.] of size l., IV.,,. denotes the multiplier associated to I.,., and 9(x) is a "square window" function defined by 9(x) = 1 if - 1/2 ~< x ~< 1/2 and 9(x) = 0 otherwise. Hence in Eq. (3) the function 9((x - x.,,,,)/1.) is equal to 1 everywhere inside I.,,. and vanishes elsewhere. The representation (3) shows explicitly that eN(x) is obtained by exponentiating a sum of square "pulses" of different sizes 1. and random amplitudes In { W..,~ }. The number of pulses of size 1, = 1/2" is _M(l,) = 2" = l,-'

(4)

The locations x,,,, of the pulses are not random, they are fixed and determined by the grid underlying the construction of the multiplicative process. It is for this reason that we refer to the random functions eu(x) as multiplicative processes on a grid, or M.P.G.'s. Multiplicative processes on a grid can also be extended to two and three dimensions, and the construction method in two dimensions is illustrated in Fig. 3. Although not quite apparent in one dimension, the grid becomes quite visible in two dimensions and gives a rather unnatural appearance to the field (Fig. 4). 2.2. Properties

Multiplicative processes on a grid (denoted by M.P.G. in the following) have been classified according to several criteria. In this section we summarise briefly this classification and list the essential properties of M.P.G.'s. Let us first introduce some necessary notations. The function ~N(x) can be regarded as a "density" function which can be used to define a measure I~N. The measure #N{ [a,b]} of any interval

A. Saucier/Physica A 226 (1996) 34 63

39

n=O

60 = 1

wl

w2

n=1

al = 1/2

w3

w4

wlwl

wlw2

w2wl

w2w2

wlw3

wlw4

w2w3

w2w4

w3wl

w3w2 w4wl

w4w2

w3w3

w3w41w4w3

w4w4

a2 = I / 4

n = 2

Fig. 3. The first two steps of construction of a deterministic two-dimensional multiplicative process. In this case the weights are not random but are fixed numbers w~. w2, w3 a n d w4 that satisfy 5"w~ = 1 if the process is conservative.

[a,b] ~ [0, 1] is simply defined by b

/~N{[a,b]} = f eu(x)dx.

(5)

a

For simplicity the total measure/~N { [0, 1] } will be denoted by ~'~N' A first classification of M.P.G.'s can be done according to the conservation properties of the total measure f2N as N increases. A M.P.G. is said to be conservative or microcanonical if for each splitting of the construction process we have W,. = 2.

(6a)

i=l

For such a process, the 2 multipliers Wi involved in a splitting remain identically distributed but are not independent of each other. Indeed, Eq. (6a) implies

A. Saucier/Physica A 226 (1996) 34-63

40

Fig. 4. An illustration of the measure generated by a 2D deterministic multiplicative process with N = 7 iterations and weights wi equal to (0.35, 0.05, 0.15, 0.45). High values of the measure are given lighter grey tones. The grid underlying the construction of the model is clearly visible and remains visible even when the multipliers Wj are random variables.

correlations between the multipliers involved in a splitting. However, multipliers belonging to different splittings or acting on different length scales remain independent and identically distributed. The conservative model is therefore a departure from the model presented in Section 2.1, where all multipliers are mutually independent. For a conservative M.P.G., the total measure (2N does not vary with N and remains equal to one. The M.P.G. presented in Section 2.1, where all multipliers are mutually independent, is said to be non-conservative or canonical. It does not respect (6a), but it respects instead the weaker constraint ( W ~ = 1.

(6b)

It should be noticed that a conservative M.P.G. always satisfies (6b), as can be seen by averaging Eq. (6a). In addition, it follows from (6a) that the weights of a conservative M.P.G. must satisfy W ~< 2. In contrast, W is not bounded for a non-conservative process.

A. Saucier/Physica A 226 (1996) 34 63

4t

For non-conservative processes f2u is a random variable that converges as N ---,~, to a limit random variable denoted by (2. Another classification of M.P.G.'s can be done according to the existence of the moments of #2. A M.P.G. is said to be convergent if (f2 q) < oo for q > 1. A M.P.G. is said to be divergent if there is a qc > 1 such that ((2 q) diverges when q > qc- The condition of convergence was established by Kahane and Peyri6re [10]: a M.P.G. is convergent if and only if Kc(q) < q - I for q > l . Conversely, i f K c ( q ) > q - 1 for q > q ~ > l then the M.P.G. is divergent. Conservative M.P.G.'s are always convergent (indeed, f2.~, is constant), but nonconservative processes can be either convergent or divergent, depending on the probability distribution of W. When analysing multifractal measures of geophysical origin, for example, the generator (if it exists) is not a quantity that is accessible directly. Consequently one usually analyses the scaling properties of spatially averaged quantities. The quantities that are typically used are the measure p(1) of an interval of size l, or else the average density ~:(/) = p(l)/1 of an interval of size 1. For multifractal measures, scaling exponents K(q) and r(q) are defined by the power law relations ({~;(/)}q) ~ 1-K(q) ,

(7a)

( { p(l) }q) ~ 1 l +~(q) '

(7b)

where r(q) is usually called the mass exponent function. The measure is said to be multifractal if (7a) (or (7b)) holds and if K(q) (or r(q)) is a nonlinear function of q. It follows from e(I) = p(l)/I and the definitions (7a, b) that the exponents r(q) and K(q) are related by K(q)=(q--

1)-r(q).

(8a)

The generalised dimensions D(q) are defined (Hentschel and Procaccia [24]) by D(q) = z(q)/(q -- 1),

(8b)

and it follows from (8a) that D(q) = 1 - K ( q ) / ( q - 1).

(8c)

For multiplicative processes on a grid, the exponents K(q) can be related to the generator K ~ ( q ) as following [8]. If the M.P.G. is conservative, then K(q) = K c ( q ) ,

(9)

for all q. If the M.P.G. is canonical and convergent, then (9) also holds for all q. Finally, if the M.P.G. is canonical and divergent, then the identity (9) holds for q ~< q~ (K(q) is not defined for q > qc). Strictly speaking, (9) holds in the limit N ~,:~. Numerical experiments show that a finite N, not necessarily very large, is sufficient for (9) to hold approximately.

A. Saucier/Physica A 226 (1996) 34 63

42

3. Disordered multifractais

3.1. Definition We shall modify multiplicative processes constructed on a grid by first randomising the locations x,.m of the pulses in eq. (3): x,,,, will be replaced by a random variable X,,m uniformly distributed on the interval [0, 1]. Furthermore, the random variables U,.,, = ln{W,,,,}, which have a probability distribution independent of n and m, will now be modulated by a scale dependent amplitude factor denoted by a(n). The sizes I, of the pulses will be discrete set of scales {1,, N1 ~< n ~< N} such that 1, decreases with increasing n (i.e. I, is not necessarily equal to 2-"). In addition, the number of pulses of size I. will be an unspecified function M(n) (i.e. not necessarily equal to 2"). The result of all these modifications is the new random function

=

N 1

The U,, mare a collection of independent and identically distributed random variables, i.e. U,,m ~ U for all (n, m). Similarly X,,m £- X for all (n, m), where X is a uniform random variable on [0, 1]. 9(x) is the square window function defined previously in Section 2. By contrast with the M.P,G. (3), the random function (10) is constructed without any grid. We shall see (Appendix A) that the absence of a grid restores the spatial homogeneity of eN(x) in a subinterval of [0, 1]. The increased randomness of (10), compared to Eq. (3), is essentially due to the random locations X,,m of the square pulses. In that sense the field (10) is more "disordered" than a multiplicative process on a grid. The main goals of this paper are first to determine the constraints that must be imposed on the various ingredients of the model (10) (i.e. the scales I, and the functions a(n) and M(n)) in order to obtain a multifractal field, and secondly to calculate the scaling properties (i.e. the generator KG(q)) of the resulting disordered multifractal.

3.2. Moment generating function of U In the course of this paper, the behavior as q ~ 0 of the moment generating function (in short M.G.F.) of U will be shown to play a significant role. The following preliminaries will consequently be useful. The M.G.F. of U is defined by ~o(q) = ( e x p ( q U ) ) ,

(11)

and will be assumed to exist for all q. In addition, we will assume that the random variable U is chosen in such a way that the M.G.F. satisfies

q2(q) ,.~ 1 + cq ~

(12)

A. S a u c i e r / P h y s i c a A 226 H996) 34 63

43

as q - , 0, where c and 7 are constants. For example, suppose that U has a Gaussian distribution of mean 0 and standard deviation o. The M.G.F. is then given by ~o(q) = exp{(q202)/2} and expansion in Taylor series a r o u n d q = 0 yields

(p(q) = 1 + 10"2q2-t-O(q3).

(13a)

C o m p a r i n g the expansion (13a) with the definition (12) then yields c = ½02 and ~ = 2. This is a special case of a more general result. Indeed, if ~0(q) is analytic at q = 0 then it can be shown that ~ot"l(0) = ( U " ) for all n ~> 0. With the assumption ( U ) = 0, the second order Taylor expansion of q0(q) at q = 0 takes also the form (13a), and therefore c = ( U 2 ) / 2 and ~ = 2 is again obtained. This result holds for all distributions having an analytic M.G.F. at q = 0 and satisfying ( U ) = 0. If ( U ) ¢ 0 and if (p(q) is analytic at q = 0, then the first order Taylor expansion of (p(q) at q = 0 yields

(p(q) = 1 + ( U ) q

+

O(q2),

(13b)

from which follows that c = ( U ) and ~ = 1. Values of ~ different from 1 or 2 can also be obtained if we consider r a n d o m variables with infinite variance. Consider for example the r a n d o m variable U = - $ 1 / 2 , where $1/2 is the stable distribution with index 1/2 (Feller [25]) that has a probability density p(s) = ( 1 / 2 x / ~ 5 ) e x p ( - l/2s) for s >~ 0 and p(s) = 0 for s < 0. In this case U takes only negative values, and both ( U ) and ( U 2) are infinite, but the M.G.F. is well defined and takes the exact form q)(q) = exp( - x ~ ) . analytic at q = 0, but it satisfies

qo(q) = 1 - xf2q 1/2 -F O(q).

This q)(q) is not

(13C)

The form (12) is therefore obtained with c = - \ , f 2 and :~ = 1/2 O t h e r examples based on stable distributions can also be found, and values of :~ between 0 and 2 can be obtained. The gaussian case examined above corresponds to the stable distribution with index 2.

3.3. A criterion of multifractality The scaling property (2b) that holds for multiplicative processes constructed on a regular grid was obtained originally by Yaglom [1,2] and M a n d e l b r o t [8]. Schertzer and Lovejoy [19] used this property as a criterion of multifractality for a b r o a d e r class of multifractal models, i.e. models that do not necessarily involve any grid. Following this idea, we shall say that the process eN(x) is asymptotically multifractal in an interval I if for any x ~ I and any real q we have In (r,~v(x)) ~ - K~(q)In (IN),

(14a)

as 1N --* O, where the generator Ko(q) is a nonlinear function of q. IN is the smallest pulse size used in the construction of eN(x). Alternatively, we will also use the following

44

A. Saucier/Physica A 226 (1996) 34-63

differential form of (14a):

(O/OIN) In (¢~(x)} ~ - K ~ ( q ) 1 ; '

(14b)

as lN ~ 0. It is stressed that (14a) is a property of the construction process of gN(x), and is not a property of the spatially averaged quantity e (/). The criterion (14a) is not to be confused with the more usual definition of multifractality expressed by Eqs. (7a, b). A field that satisfies the criteria of multifractality (14a, b) will not necessarily satisfy the normalisation condition K~(1) = 0. However, the field can always be normalised a posteriori by defining a normalised field

E*(x) = ~,N(X)/ (~N(X) 5.

(15)

If (14a) holds, then it is easily shown that e*(x) satisfies In ([e,*(x)] q) ~ - [ K G ( q ) -- qKG(1)] In(IN)

(16a)

as lN ---,0. Comparing (14a) and (16a), we conclude that the generator of the normalised field is / ( * ( q ) = Ko(q) - q / ( ~ ( 1 ) ,

(16b)

which always satisfies the normalisation condition K * ( 1 ) = 0. This normalisation procedure will be used systematically in this paper.

4. Disordered multifractals with privileged scale ratios

4.1. Condition of multifractality and resulting generators" In this section, we make a choice of length scales I, identical to the scales used in a multiplicative process on a grid, i.e. I, = 2 - " ,

(17)

where 2 > 1 and n >~ N~ are integers. The construction of the random field eu therefore involves a privileged scale ratio 2, but no grid. Our problem is to determine the conditions to impose on the functions M(n) and a(n) in order to obtain a multifractal field, and then to derive the corresponding generator. The first step is to calculate the asymptotic form of (~/c~lu) ln((e~(x) )) as lu --* 0. It is shown in the appendix A that if x lies in the interval IN,/2 < x < 1 -- lu,/2,

(18a)

where IN~ < 1/2, then

On(Is) M(lu)ln[lu~o(qfi(lu)) + 1 - Is]

~?01s(In (e~(x)) ~ - ~ u

(lSb)

A. Saucier/Physica A 226 (1996) 34 63

45

holds for N large enough, or equivalently for lN small enough. Notice that (~/~.IN) In (~:q(x)) does not depend on x because eN(x) is spatially homogeneous in the interval (18a). The function n(1) = - l o g ; ( / ) in Eq. (18b) is obtained from Eq. (17) by expressing n in terms of l,, whereas IQ and a are the functions M ( n ) and a(n) expressed as a function of l,, i.e. JQ(l) = M ( n ( l ) ) and a(l) = a(n(I)). Let us now assume that the function ~(l) decreases as I decreases, and moreover that ~(I) --+ 0

as 1--+0.

(19)

This amounts to giving smaller amplitudes to pulses having a smaller size. In order to use the condition of multifractality (14b), we consider the asymptotic form of i 18b) in tile limit IN --+ 0. As lN --+ 0, the argument q~(l~,) of the function q~ goes to zero for any fixed q, and consequently the asymptotic property (p(q) ~ 1 + cq ~ as q -~ 0 (as discussed in Section 3.2) can be used. Using in addition the asymptotic property In(1 + x) ~ x as x --+ 0 yields the asymptotic form f-- In (~:q(x)) t?lN

C

.... q~l(Iu)~(IN) ln~a~

(20)

as IN--+ 0. It is clear that Eq. (20) will satisfy the condition of multifractality (14b} if

holds for all I, where G is a positive constant. Eq. (21) is therefore the condition of multifractality when 1, = k-" and a(I)--, 0 as 1--, 0. The multifractality of disordered multifractals is therefore achieved via a balance between a number of pulses M(I) and the amplitude factor a(/). Assuming that the condition of multifractality (21) is satisfied, it follows immediately that the generator of this multifractal process is K~(q) = [ G c / l n ( 2 ) ] q ~ ,

{22a)

and the generator of the normalized process (as discussed in section 3.3) is therefore K * ( q ) = [Gc/ln(2)](q ~ - q).

(22b)

The main lesson to be drawn from the result (22b) is that these disordered multifractals (i.e. I, = 2 " and a(l)--+ 0 as l ~ 0) exhibit a "degeneracy" of their scaling properties with respect to the random variable U. By degeneracy we mean that different random variables U can have exactly the same generator. Indeed, according to (22b) the generator K ~ ( q ) depends only on the parameters ~ and c, which do not determine uniquely the probability distribution of U. This implies in particular that all probability distributions having an analytic moment generating functions and satisfying ( U 2) = o"2 and ( U ) = 0 will have exactly the same generator, with c lo'2 and ~ = 2 (as long as 2 and G are constant). This phenomenon of degeneracy is a major difference between this disordered multifractal and multifractals on grids, for which the generator depends on the whole probability distribution as seen from Eq. (2c). Notice also that according to Eq. (22b) the generator K~,(q) vanishes for =

A. Saucier/Physica A 226 (1996) 34 63

46

all q when ~ = 1. This implies for example that K ~ ( q ) will vanish whenever ( U ) :~ 0, as long as q~(q) is analytic at q = 0. In such a case the field e.N(x) is similar to a white noise and is therefore not really multifractal. For any multifractal field characterized by a generator K ~ ( q ) we can define a codimension function by C6(q) = K ~ ( q ) / ( q - 1). For degenerate generators of the form K ~ ( q ) = k(q ~ - q), where k is a constant, it can be shown that CG(1)= limos1 C6(q) = k(ot - 1), as long as c~ ¢ l. It follows that the codimension function can be written in the form Cc(q)=

Cc(1) q~ - q

~ _ l q - I

(22c)

The family of functions (22c) has been proposed by Schertzer and Lovejoy [19, 20] as universal codimension functions for various geophysical fields. From this point of view, two parameters C6(1) and ~ are sufficient to characterize the generator of a multifractal field. Let us now examine another type of variation for the function 5(1). We shall simply assume that the pulses have the same amplitude for all pulse sizes, i.e. that (23)

5(1) = a

for all l, where a is a constant. This is the case for example for all multiplicative processes on a grid, as seen from Eq. (3). In order to use the multifractality condition (14b), we consider again the asymptotic form of (18b) as lu--* 0. Replacing 5(1) by a and using the asymptotic property ln(1 + x) ~ x as x --* 0 yields the asymptotic form

(31N

In (e.~(x))

1 [qo(aq) - 1]Al(/u), ln(2)

(24)

as lu --* 0. It follows that the condition of multifractality (14b) will be satisfied if M ( I ) = G'/1

(25)

holds from all l, where G' is a positive constant. (25) is the multifractality condition that must be satisfied for disordered multifractals such that I, = 2-" and 5(1) = a for all I. Notice that the condition (25) is satisfied by multiplicative processes on a grid, as seen from Eq. (4). Notice also that (25) is a special case of (21). Assuming now that (25) is satisfied, it follows immediately from (24) and (14b) that the generator takes the form K ~ ( q ) = [G'/ln(2)] [q~(aq)- 1],

(26a)

and the generator of the normalized field becomes K ~ ( q ) = [G'/ln(2)] { [q~(aq)- 1] - q E c p ( a ) - 1]}.

(26b)

In contrast with the previous case, we obtain now a generator that depends on the whole probability distribution of U via the moment generating function q~(q). In other

A. Saucier,,'Physica A 226 ¢1996) 34-63

47

words, there is no degeneracy of the generator with respect to the random variable U. This shows that a randomization of pulse locations does not necessarily imply a degeneracy of the scaling properties of the field. 4.2. E x a m p l e s

It is instructive to examine what type of functions can satisfy the multifractality condition (21). There are infinitely many ways of choosing two functions/~(l) and g(l) such that (21) will be satisfied. If we choose to focus on power law solutions, we can introduce as an example the family of solutions fi(I) = ~rl Id- 1)/~,

(27a)

M(I) = M l l -a ,

(27b)

where d ~> 1,/~1 > 0 and a > 0 are real parameters. For d > 1, then a(l) --* 0 as l ~ 0 and the generator is given by Eq. (22b) with G = M l a ~. For d = 1, then a(l) = a for all 1 and the generator is given by (26b) with G' = .Q1. Notice that for d > 1 tile generator is independent of d. As an illustration of the above models, we constructed a disordered multifractal using Eq. (27a, b) with a = 1, . ~ = 1, d = 1 and a random variable U uniform on [ - 1/2, 1/2]. The scales are l, = 2-" (the privileged scale ratio is 2) with N = 12 and N~ = 3. The size of the largest pulse is therefore 1/23 and the smallest pulse has a size 1/'212. The moment generating function of U takes the form q~tq) = (e q/~- - e - ~ ' 2 ) / q . Expanding the exponentials in Taylor series around 0 yields for ~o(q) the expansion ~o(q) = 1 + (1/3!)(q/2) z + O(q 4) and therefore c = 1./24 and ~ = 2. The field t:N(x) was constructed by adding one by one the contributions of each pulse to the function In ru(X), that was sampled with a resolution of(l/10)th of the size of the smallest pulse. Exponentiating this sum yields the final field, which was sampled only in the interval (18a) where the field is multifractal and spatially homogeneous. A picture of a typical realisation is displayed in Fig. 5. The singular nature of this function is clearly seen from the large and localised singularities. According to equation (26b) the generator of this multifractal takes the form K * ( q ) = (1/ln (2)){[-~0(q) - 1] - q[-q~(1) - 1]}.

(27c)

The scaling properties of this realisation were examined by calculating the generating function Zq(6) = ( 6 / L ) q- l (Pqx((J) ) s / [ (Px(~J) ) s ] q ,

(27d)

where ( ..- )s denotes a spatial average and L is the sample size. According to Eq. (7b}, the generating function (27d) satisfies Zq(6) ~ 6 ~!ql for large enough samples. Z2(6) was plotted as a function of 6 for q = 2 in Fig. 6. The scaling is quite good if 1N ~ 6 ~ lx,. Choosing an interval where the scaling appears to be best, we obtained by linear regression the exponents r(q) and calculated from them the exponents K ( q ) with

A. Saucier/PhysicaA 226 (1996) 34 63

48

al Fig. 5. A realisation of a disordered multifractal in one dimension. The length scales are I, = 2 ", the amplitude factor is unity for all pulses and there are 1,~-1pulses of size /,. U is uniform on [ 1/2, 1/2]. N l = 3 a n d N = 12.

Eq. (8a). We then plotted both the generator K~(q) calculated with (27c) and the estimated K(q) on the same plot in Fig. 7. The agreement between K(q) and K~(q) is excellent for 0 < q < 1.5, by K(q) is a little below K*(q) for q > 1.5. A perfect agreement for all q cannot be expected from a single realisation because K(q) depends on rare events for q large. Averaging over m a n y realisations is necessary to capture accurately the scaling properties of such events. The excellent agreement between the generator and the K(q) measured on a single realisation strongly suggests that the identity K(q) = K~(q) can also hold for disordered multifractals.

4.3. Power spectrum ofln eN(x) In this section we examine the p o w e r spectrum of the field In eN(x) for disordered multifractals with privileged scale ratios. It is shown in the Appendix B that for a disordered multifractal of the form (10), and in the special case where ( U ) = 0, the power spectrum of In eN(x) can be written in the a p p r o x i m a t e integral form l,v

P(k) ~

4 ( U 2) f

k2

On

dl ff[ ]VI(1)~2(1)sin2(kl/2),

(28)

if I1 is small enough. F o r disordered multifractals with privileged scale ratios, it was shown in Section 4.1 that the multifractality condition imposes a well defined form for the p r o d u c t ]fl(l)M(1). It follows that if ~ = 2, then the power spectrum (28) can be

A. Saucier/Physica A 226 (I996) 34 63 0

.......

t

,i

........

49

!

.....

ol -2 -3 .4

m

s,z. o, ,..

-5

_ ,,./..

i m.'...

-6

L

...................:

............

-7 -8 1 O" 4

0.001

0,01

0.1

1

5 Fig. 6. Scaling of Zz(6) for the disordered mullifractal displayed in Fig. 5. The power law behavior is generally fairly good but is best in the central range.

0.4

......

0.35

I

....

0,3 0.25 A O" v,v,

"

0.2 0.15 0.1 0.05

~._.~

0

.

-0.05 0

.

.

.

4~+ 0.5

1

1.5

2

2.5

3

0

Fig. 7. The solid line is the generator K~(q) predicted theoretically in the paper, while the dots represent the K(q) measured on the disordered multifractal of Fig. 5. The agreement between both is excellent for q < 1.5. The difference between K(q) and K~(q) is due to statistical errors.

calculated without making any additional assumptions about the functions M(1) and 8(1). We shall only consider the case ~ = 2 for which no assumptions are required but the multifractality of the field. Using in Eq. (28) the i d e n t i t i e s / ~ ( l ) 8 2 ( / ) = G/l (condition (21)) and ?n/?l = -1/(ln(2) I), and then making the change of variable t = kl yields kll

P(k) ~

4(U2>G f klog(2~

kl~

dtt-2sin2(t/2)"

129a)

A. Saucier/Physica A 226 (1996) 34-63

50

Using integration by parts, (29a) can be shown to take the exact form P(k) - 2(U2)Gk21082[_\[(C°s(k/X)ll- 1

cos(kiN)is - 1)

4- k [Si(kl 1) - Si(ktN)]l ,

(29b)

where Si(z) = So sin(t)t- 1 dt is the sine integral function. If we focus on the range of wave numbers 2~/11 <~ k < 2~/lN, then the spectrum (29b) takes the approximate form P(k) ~

2(U2)G /log-----~ [Si(kll

- S~(klN)] .

(29c)

In Eq. (29c), the prefactor Si(kll) - Si(klN) oscillates around a mean that varies slowly compared to k -1. Hence we conclude that lneN(x) is a l / f noise when ( U > = 0 and --2. The same result holds for continuous cascades [19,20], for which lneN(X) is a l / f noise for any ~.

5. Disordered multifractals without privileged scale ratio 5.1. Condition o f multifractality and resulting generators

We consider this time a set of length scales I, that does not involve any privileged scale ratio. We choose a discrete set of scales identical to the one used in Fourier series, i.e. l, = 1/n,

n ~ N1.

(30)

The construction of this new field eN involves neither a grid (the multifractal is therefore "disordered") nor a privileged scale ratio 2. Our problem is again to determine the conditions to impose on the functions M(l) and ~i(l) in order to obtain a multifractal field, and to determine the corresponding generators. It follows from 1 = 1/n that ~n/~l = - 1 - 2 and Eq. (18b) then takes the form Ol~ In (e~,(x) > ~ - l~ 2/~ (l N) In [1N~o(qa(lN)) + 1 -- 1N] .

(31a)

In order to use the multifractality criterion (14b), we consider again the asymptotic form of (31a) in the limit lN ---) O. Assuming again that t~(1) ~ 0 as l --) 0, we show as previously that ~l~ In (e~(x)) ~ -- cq ~l~, 1 M(lN)a~(lN),

(31 b)

A. Saucier/Physica A 226 (1996) 34 63

51

as IN ~ 0 . It follows from Eq. (31b) that the field will satisfy the multifractality condition (14b) if (32)

~l(I)~(1) = G,

for all l, where G is a positive constant. Eq. (32) is the new condition of multifractality when I, = 1/n and d(1) --. 0 as I --* 0. This condition is different from the previous one (21), obtained for I, = 2-", which shows that the condition of multifractality depends on the set of length scales {l~, 12, ..., lu}. Assuming that the condition of multifractality (32) holds, it follows immediately from (31b) that the generator takes the form K~(q) = cGq ~, from which follows (33)

K ~ ( q ) = cG(q ~ -- q) ,

i.e., the scaling is degenerate. If we assume now that ~i(I) is constant, i.e. that d(I) = ~r for all l, then (31a) takes the asymptotic form ~ ? In (eq(x)) ~ - [g0(aq) - 1] l~ ~-M(IN), ?IN

(34)

as lu --* 0. The condition of multifractality is therefore M(t) = G',

(35)

where G' is a positive constant. Eq. (35) is a special case of(32) obtained when ci(l) is constant. The generator that corresponds to (35) is KG(q) = G ' [ q ~ ( a q ) - 1], from which follows Kg(q) = G'{qo(aq)- 1 - q E q ~ ( a ) -

1]},

(36)

i.e. the scaling is not degenerate. Solutions of Eq. (32) are for instance the family of power law solutions M(1) = M1 l - d ,

(37a)

~(l) = al a/~ ,

(37b)

where cr > 0, )~ra > 0 and d ~> 0 are real parameters, and the constant G then takes the form G = M ~ a ~. Of course, m a n y other solutions can be found. I f d > 0, then 8(l) ~ 0 as l -~ 0 and the generator is given by (33) and is degenerate. If d = 0, the scaling of the generator is given by (36) wih G' = Mt and is not degenerate. This is a case where both the amplitude 8(1) and the number of pulses M(1) are constant. 5.2. P o w e r spectrum o f l n e N ( x )

As previously, the power spectrum of In e,N(x) can be computed without making specific assumptions on the functions g(1) and M(l) as long as ( U ) = 0 and ~ = 2.

A. Saucier/Physica A 226 H996) 34-63

52

R e p l a c i n g ct = 2 a n d ~n/Ol = - l - 2

in t h e i n t e g r a l (28) a n d u s i n g t h e c o n d i t i o n o f

m u l t i f r a c t a l i t y (32), the c h a n g e o f v a r i a b l e t = kl yields kin

P(k) ~ 4 ( U 2k) G

f

dtt-2sin2(t/2),

(38)

kl~

w h i c h is of t h e f o r m (29a), a n d t h e r e f o r e In eN(x) is a g a i n a 1 / f noise.

6. Generalisation to two dimensions A 2 - D e x t e n s i o n of t h e p r e v i o u s m o d e l (10) c a n be c o n s t r u c t e d by e x p o n e n t i a t i n g sums of cylindrical or square I-0,1] × [ 0 , 1 ] .

Introducing

the

pulses r a n d o m l y notation

l o c a t e d w i t h i n the u n i t s q u a r e

x = (x,y)

and

X,,m = ( X . . . . Y,.m),

a

(a) Fig. 8. (a) A 2-dimensional disordered multifractal constructed with 1, = 2 ", M(/) 1/I 2 and 5 = 1. We used N = 8 and N1 = 3, so that the largest pulse has a size of 1/23 and the smallest 1/28 (the whole image has a size unity). Large values are given lighter grey tones. In some places, and in particular in the center, the shape of the circular pulses used in the construction is still visible. These circles appear as light or dark disks in the image. (b) We examine the same field but this time we focus only on the largest singularities by giving a black tone to most small values. =

A. Saucier/Physica A 226 (1996) 34 63

53

(b)

Fig. 8b. (Continued). multifractal sum of pulse in two dimensions can be written in the form

~ M(n) ~. a(n)U.,m .q[(x -- X..m)/l.]

~:N(X) = exp n=NI

m=l

) .

(39)

F o r a cylindrical pulse the function 9 is defined by 9(x) = 1 if []x ]] ~< 1/2 and 9(x) = 0 if Hxl[ > 1/2. The region where 9(x) = 1 is a disk of diameter 1 centered on the origin, and therefore 9[(x-X.,m)/1.] is a cylindrical pulse of diameter I. and amplitude 1 centered a b o u t the point X.,m. X.,m and Y..m are independent r a n d o m variables uniformly distributed on [0, 1], and U.,m is a r a n d o m variable with well defined m o m e n t generating function (p(q). It is straightforward to generalise the previous calculations to two dimensions (appendix C). We obtain in particular

g'l~ In (~:q(x)) ~ On(Is) ~lu f4(lN)ln[fl2N~P(qa(l'~'))+ l - - f l ' ~ ] '

(40)

as /N --+ 0, where f = ~/4 if the pulse is cylindrical and f = 1 for a square pulse. Proceeding as previously, it is easy to show that the conditions of multifractality take

A. Saucier/Physica A 226 (1996) 34-63

54 the form

]~/I(l)a~(l) = G / l z , I~4(1)~'(l) = G / l ,

I, = 2 - "

(41a)

I. = 1/n

(41b)

(G is a p o s i t i v e c o n s t a n t ) . E q s . (41a, b) h o l d f o r b o t h c a s e s ~ ( l ) = a a n d ~i(l)--. 0 as l --* 0. T h e g e n e r a t o r s a r e t h e s a m e as p r e v i o u s l y , i.e. t h e y a r e d e t e r m i n e d o n l y b y t h e scales I, a n d t h e b e h a v i o r o f t h e f u n c t i o n ~(l). As an illustration of these 2D disordered multifractals, we have constructed two s u c h fields. I n b o t h c a s e s t h e r a n d o m

variable U was chosen to be uniform on

(a) Fig. 9. (a) A 2-dimensional disordered multiffactal constructed with /. = 1/n, IQ(I) = IQ1/I , -M1 = 13, N = 256, N1 = 2 and ~i = 1. Large values are given lighter grey tones. The scales I, are much closer to each other and the field is visually more "nebulous". (b) We examine the same field but this time we focus only on the largest singularities by giving a blue color to most small values, and light pink colors to higher values (the lighter the larger). The support of these large singularities has a dimension lower than two and may have a resemblance with smoke filaments, clouds or nebulae.

A. Saucier/Physica A 226 (1996) 34 63

55

(b) Fig. 9b. (Continued).

[--- 1/2, 1/2]. The first disordered multifractal is constructed with I, = 2 ", ]~(l) = l/l 2 and d = 1 so that the multifractality condition (41a) is respected. We used N = 8 and N~ = 3, so that the largest pulse has a size of 1/23 and the smallest 1/28. The field was again constructed by adding one by one the contributions of each pulse to In e,N(x), and then exponentiating. The function was sampled with a resolution of one half the size of the smallest pulse and displayed on a 512 × 512 grid. A grey-shade representation of the field is given in Fig. 8a, b. In some places, and in particular in the center, the shape of the cylindrical pulses used in the construction is still visible. These pulses a p p e a r as light or dark disks in the image. The second field was constructed with the same m e t h o d but this time with the scales I, = 1/n, ~1(1) = ~l~/1, M~ = 13, N = 256, N1 = 2 and d = 1, so that the multifractality condition (41b) is satisfied. The resulting field was plotted in Fig. 9a. The scales I, are m u c h closer to each other and the field is visually m o r e "nebulous". In Fig. 9b we looked at the same field but this time we focused on the largest singularities by giving a blue color to a large p r o p o r t i o n of the

56

A. Saucier/Ph.vsica A 226 (1996) 34 63

small values of the field, and light pink colors to other values. The support of these singularities has a dimension lower than 2 and may have a resemblance with smoke filaments, clouds or nebulae.

7. Summary and conclusions In this paper we have introduced a new family of spatially homogeneous 2 multifractals. The models considered are constructed by exponentiating a sum of pulses of various sizes and amplitudes. By contrast with multifractals constructed on a regular grid, the locations of the pulses are random and mutually independent. In that sense, these multifractals are more "disordered". The absence of a grid in their construction also insures spatial homogeneity. In the context of turbulence, where cascade processes have been proposed to model the energy cascade from large to small scales, the multiplicative process is thought to simulate an energy flux from a "mother eddy" to "daughter eddies". If the process occurs on a grid, daughter eddies are always contained inside their mother eddy. For disordered multifractals the spatial positions of large and smaller scale eddies are random and unrelated (i.e. independent), and in that sense the relationship between the positions of mother and daughter eddies has been broken. The general goal of this paper was to study the effect of this randomization on the scaling properties of the field. In order to define our model of disordered multifractal, several ingredients must be specified: a discrete set of length scales {11,12, . . . , IN} corresponding to the pulse sizes, the number M(1,) of pulses of size 1, and an amplitude factor ~(1,). The amplitude of the mth pulse of size l, takes the form ~i(l,)U,, m, where the random variables U,.,, are mutually independent and identically distributed. It is also assumed that the moment generating function ~p(q) of U,.m is well defined and satisfies ~o(q) ~ 1 + cq ~,

as q --* 0.

(42)

Our first task was to determine the constraints to apply on the functions M(I,) and t~(l,) to obtain a multifractal field. It was found that multifractality is achieved via a balance between the number of pulses and the amplitude factor. More precisely, it was shown that the field is multifractal if ~ 4 ( l ) ~ ( 1 ) = B(l, D ) ,

(43)

where the function B(I, D) depends on the set of length scales {/a, 12, .-., 1N) chosen for the construction, D is the dimension of space (both D = 1 and 2 were examined in this paper) and ~ is defined by Eq. (42). In particular we found that if we choose the length

2I.e. all statistics,e.g. probabilitydistribution and autocorrelationfunction,are invariant under translation.

A. Saucier/Physica A 226 (1996) 34 63

57

scales I, = ).-", N1 ~< n ~< N, where 2 is an integer scale ratio, then B(l, D) = G/l D ,

(44a)

where G is a positive constant. However, if we choose instead the scales /,, = l/n, identical to the scales used in Fourier series, then we get B(l, D) = G/l D- 1 ,

144b)

where G is again a positive constant. It is therefore clear that an infinite number of solutions M(l) and cT(I) can be found to satisfy the multifractality condition (43). For example, in one dimension and for I, = I/n, the solutions ]~(l) = M , l a,

(45a)

~(l) = ~l e/~ ,

(45b)

where cr > 0, M~ > 0 and d ~> 0 are real parameters, satisfy the multifractality condition (43) and the constant G then takes the form G = MI a ~. There are many ways to achieve multifractality in the context of disordered multifractals, and we may therefore conclude in particular that a relationship between the locations of mother and daughter eddies (i.e. pulses) is certainly not necessary to achieve multifractal scaling. Assuming that the multifractality condition (43) was respected, we examined the scaling properties of the field, which are specified by the generator K*(q). It was found that two different types of scaling exponents could be obtained, depending on the behavior of the function ~(I). The first type of scaling occurs when ~i(l)---, 0 as l ~ 0. In that case, it turns out that for ~ ¢ 1 the generator can be written in the form (46a) where CG(1) is a constant that depends on c, as defined by Eq. (42). This result shows that these disordered multifractals exhibit a "degeneracy" of their scaling properties with respect to the r a n d o m variable U. By degeneracy we mean that different random variables U can produce exactly the same generator. Indeed, K * ( q ) depends only on the two parameters c and :~ which do not determine U uniquely (Eq. (42)). This implies for example that if ~o(q) is analytic at q -- 0 and ( U ) = 0, then K ~ ( q ) depends only on the variance of U. The same function (46a) was derived earlier by Schertzer and Lovejoy [19, 20], but using a quite different gridless model (continuous cascades). The fact that we obtained the same result with a completely different model is interesting because it gives additional support to the notion of universality in fully developed turbulence. A high degree of randomness and disorder in high Reynolds number flows might result in universal scaling laws analogous to the famous k 5/3 energy spectrum (Kolmogorov [-27]).

A. S a u c i e r / P h y s i c a A 2 2 6 (1996) 3 4 - 6 3

58

A second type of scaling behavior is obtained when a(l) = a for all I. In that case, it was shown that the generator takes the form

K~(q) = C{~o(aq)- 1 - q [ q ~ ( a ) - 1]},

(46b)

where C is a constant. In contrast with the previous case, we obtain here a generator that depends on the whole probability distribution of U via the moment generating function q~(q) of U. This shows that a randomization of the pulse locations does not necessarily lead to a degeneracy of the scaling properties of the field. The scaling behavior (46b) cannot be obtained in the context of the continuous cascades introduced by Schertzer and Lovejoy [19,20]. Finally, assuming only that the multifractality conditions (43) holds, it was shown in one dimension that the field In eN(X) is always a l / f noise when ( U ) = 0 and ~ = 2. A similar result holds for the model of continuous cascades [19,20], for which the logarithm of the field is a l / f noise for any ~.

Acknowledgements I thank Jiri Muller for stimulating discussions, encouragement and for his careful editing of the manuscript.

Appendix A. Calculation of In( [~N(X)] q > in one dimension Using the mutual independence of the random variables involved in the expression (10) of gN(X), we can first average [eN(x)] q (the average of products of independent random variables equals the product of the averages) and then take the logarithm to obtain N

ln(([gN(x)]q)) =

~, M(n)ln((exp{q a(n)Ug[(x - X ) / l , ] } ) ) .

(A.1)

n ~ N1

Consider first the average quantity Y, = (exp{q a(n)Ug[(x - X ) / I , ] } ) .

(A.2a)

For simplicity, we denote 0 - qa(n)U; X being uniform on [0, 1], averaging first on X yields 1

(A.2b) 6

A. Saucier/Physica A 226 (1996) 34 63

59

where the symmetry 9(x) = 9 ( - x ) of the square pulse 9(x) was utilized. 9 [(t - x)/I,] is a square pulse of size I, centered about the point x. If we assume that (A.2c)

I,/2 < x < 1 - 1,/2,

then the integral (A.2b) takes the exact form (A.2d)

Y, = ( l , e ° + 1 - I , ) .

Denoting the m o m e n t generating function of U by ~p(q) = ( e x p { q U } ) and remembering that 0 - q a ( n ) U , Y, becomes (A.2ej

Y, = l, tp(qa(n)) + 1 - I,.

Eq. (A.2e) holds for all n only if the condition (A.2c) is satisfied for all n, i.e. as long as (A.3)

Ix,~2 < x < 1 - Iu,/2 ,

where IN, is the m a x i m u m value of I, (indeed we have assumed that I, decreases with increasing n). In the following, it will be assumed that the condition (A.3) is satisfied. Replacing (A.2e) in the expression (A.1) yields finally an exact expression for In ( [~:N(X)]" 5: N

ln(([e,N(x)]q)) =

~

M(n)ln[l,+p(qa(n))

+ 1 - l,].

(A.4!

tl - - N t

It will be useful to introduce an approximate integral form for the discrete summation (A.4). Such an approximation is possible if the scales I, are close enough to each other. Let us therefore assume that the scales 1, have been chosen in such a way that the differences 1,, - I n + l decrease with increasing n, e.g. 1, = 2 - " or I, = 1In. For n large enough, say n > / N 2 ~ NI, the differences I, - 1,+1 become small enough to justify an integral approximation of the summation (A.4). Choosing I, as a variable of integration and introducing the notations M ( n ) = ~1(1,) and a(n) = ~(l,), then (A.4) can be written in the approximate form N2

ln(([r,u(X)]q)) ~

~ r/

M(n)ln[l,+p(qa(n))

+ 1 -- l,]

N 1

+

dl~

M(l)ln[l~o(q~(l))

+ 1 - l].

(A.5)

Differentiating (A.5) with respect to IN yields /~1~ ln([~N(x)]q~ ~ ¢?n(1N) ~?1--~-~IVI(IN)ln[INq~(q~(lN)) + 1 -- IN]

(A.6)

that holds approximately if N is large enough, or equivalently if IN is small enough.

A. Saucier/Physica A 226 (1996) 34-63

60

Appendix B. Power spectrum of ln[~N(X)] in one dimension The power spectrum of the random function YN(x)= ln[eN(x)] is defined by P(k) - (If(k)12), w h e r e f ( k ) is the Fourier transform of Yu(x). Using Eq. (10) the linearity of the Fourier transform yields

f(k) = ~ ~ a(n)U.,mI.,m(k), n

(B.1)

m

where I..m(k) is the Fourier transform of the square pulse, i.e.

l.,m(k) =

i

eikxg[(X -- X.,m)/l.] dx = (2/k)sin(kl,/2)exp(ikX.,m).

(B.2)

Replacing (B.2) in Eq. (B.1) leads to

f (k ) = (2/k)~ ~ a(n) U., msin(kl./2) exp(ikX.,m) . n

(B.3)

m

The square modulus o f f ( k ) isf(k)f*(k) and using Eq. (B.3) it takes the form

(4/k 2 ) Z n

~ a(n)a(n') U.. mU., ~, sin( kl./2) sin(kl., /2 )

Z 2 n' m m'

× exp[ik(X.,m - X.'m')]. We shall make now the additional assumption ( U . , , . ) = 0. Since the variables U.,,. are mutually independent, averaging the above sum leaves only the terms with n = n' and m = m' (the other terms vanish because (U.,m) = 0), and therefore we get

P(k) = ( 4 ( u z ) / k 2 ) ~ ~ aZ(n) sinZ(kl./2). n

(B.4a)

m

The argument of the summation being independent of m, we finally obtain the exact result N

P(k) = ( 4 ( u z ) / k 2) Y" M(n)ae(n)sinZ(kl./2).

(B.4b)

n = NI

The form (B.4b) of the power spectrum is valid for all models of disordered multifractals such that ( U ) = 0 and c~ = 2. It will be useful to write an approximate integral form of this power spectrum. Using the same notations as previously and choosing I. as the variable of integration leads to Ix

P(k) ~ ( 4 ( g 2 )/k 2)

d l ~ _~I(l)Se(l)sin2(kl/2), Ii

that holds if 11 is small enough.

(B.4c)

A. Saucier/Physica A 226 (1996) 34 63

61

Appendix C. Calculation of In ( [~N(X)]q) in two dimensions In two dimensions the field is given by ~:N(x) = exp

~ II =

~, a ( n ) U .... g [ ( x - X , . m ) l / l , ]

,

tC.I)

i~" 1 m = l

where x = (x, y). We have X,, ~ __aX for all (n, m), with X = (X, Y ), where X and Y are independent and uniformly distributed on [0, 1]. 9(x) is a cylindrical pulse of diameter 1 and height 1 centered at the origin, i.e. g ( x ) = 1 if Hxll ~< 1/2 and g ( x ) = 0 if ]lx][ > 1/2. Proceeding as in Appendix A, we obtain N

ln{ < [Z,c(x)] q >) =

~ I1

M(n) ln(
(C.2)

N I

We consider first the average quantity Y,, - ( e x p { q a i n l U g [ I x

(C.3a)

- X)/I,,] ] 5.

F o r simplicity, we denote 0 - qa(n)U. X and Y being independent and uniform on [0, 1], (C.3a) takes the form 1

o

1

o

/

where the radial s y m m e t r y g(x, y) = g ( - x , -3') of the cylindrical pulse was utilized. g[(~ - x)/l,,(fl - y)/I,] is a cylindrical pulse of diameter I, centered a b o u t the point (x,y). If we assume that both x and y satisfy 1,/2 < x < 1 - 1,/2,

1,/2 < 3' < 1 - 1,,/2,

{C.3c)

then the cylindrical pulse remains entirely inside the unit square [0, 1] x [0, 1] and the integral (C.3b) takes the exact and simple form Y,.

=

(fl 2 e ° + 1 -f

12 ,

(C.3d)

5,

where.[ = 7r/4 is a factor that comes from the area 7r12/4 of a disk of d i a m e t e r / , . If we use a square pulse of size 1 and amplitude 1, instead of a cylindrical pulse, Y,, keeps exactly the same form but f = 1. D e n o t i n g the m o m e n t generating function of U by (p(q) = ( e x p ( q U ) ) and r e m e m b e r i n g that 0 = qa(n)U, Y, becomes Y, =.f12, ~o(qa(n)) + 1 - f l 2 .

(C.3e)

Eq. (C.3e) holds for all n only if the condition (C.3c) holds for all n, i.e. as long as 1~.,/2 < x < 1 - lx1/2,

l~r/2 < y < 1 -- 1,~,/2,

(C.4)

A. Saucier/Physica A 226 (1996) 34-63

62

where IN1 is the maximum value of I, (indeed we have assumed that l. decreases with increasing n). In the following, it will be assumed that the condition (C.4) is satisfied. Replacing (C.3e) in the expression (C.2) yields finally an exact expression for ln([~N(x)]q>: N

M(n)ln[flE~o(qa(n)) + 1 - f l 2 ] .

ln(([e,N(x)]q)) = ~

(C.5)

n -- N x

Proceeding as previously in Appendix A, we introduce an approximate integral form for the discrete summation (C.4) and obtain N2

M(n)ln[flEq~(qa(n)) + 1 --fl, 2]

ln(([gN(x)]q)) ,,~ ~ n -- N I

+

dl-~ M(l)ln[fl2~o(qg~(l)) + 1 - f l 2 ] .

(C.6)

IN 2

Differentiating (C.6) with respect to lN yields ~? ln(([eN(x)]q)) ~

On(IN)

MI(lN))ln[fl~o(qg~(lN)) + 1 - fl~]

(C.7)

that holds approximately if N is large enough, or equivalently if lu is small enough.

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